Differential Geometry

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By Cao H. D., Chow B., Chu S. C., Yau S. T. (eds.)

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3) in a full neighborhood of Wo E IR? The standard interior regularity theory (cf. in particular Ladyshenskaya - Ural'ceva [1, p. p now enables us to bound the second derivatives of Y -and hence of X - in L in a neighborhood of Wo in terms of the Dirichlet integral of X and its modulus of continuity. 3) which is normalized by a three-point-condition. 2-bound for X implies a bound for V X in LP, V P < 00. P, V P < 00. In particular, V X E C a , Va < 1. 2), cf. g. 30]. 5. Now we direct our attention to the regularity of the parametrized surface.

The remarks by Hildebrandt [4, p. J) As we shall see, by a simple variation of the classical concepts of Ljusternik-Schnirelman, resp. Palais and Smale the Plateau problem can be naturally incorporated in the frame of these methods. This extension of Ljusternik-Schnirelmann theory and its application to the Plateau problem was presented in Struwe [1]. In abstract terms we may regard this method as an extension of Ljusternik-Schnirelman theory to functionals defined on closed convex sets of (affine) Banach spaces and satisfying a variant of the Palais - Smale condition.

Uf3,p = {z E Mllz - yl < P for some y E Kf3},p >0 constitute fundamental systems of neighborhoods of Kf3. Proof: By continuity of g, clearly each Nf3,6 and each Uf3,p is a neighborhood of Kf3. Hence it remains to show that any neighborhood N of Kf3 contains at least one of the sets N f3 ,6, Uf3 ,p. Suppose by contradiction that for some neighborhood N of Kf3 and any o > 0 we have N f3 ,6 q. N. Then for a sequence Om -+ 0 there exist elements Zm E Nf3,6m \N. ) the sequence {zm} accumulates at a critical point Z E Kf3.

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Collected papers on Ricci flow by Cao H. D., Chow B., Chu S. C., Yau S. T. (eds.)

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